The degenerations between the (finite-dimensional) algebras from a certain variety \(\mathfrak {V}\)  defined by a set of identities have been actively studied in the past decade.
The description of all degenerations allows one to find the so-called rigid algebras and families of algebras, i.e. those whose orbit closures under the action of the general linear group form irreducible components of \(\mathfrak {V}\) 
(with respect to the Zariski topology).
We list here some works in which the rigid algebras of the varieties of
all 4-dimensional Leibniz algebras,
all 4-dimensional nilpotent terminal algebras [1]},
all 4-dimensional nilpotent commutative algebras,
all 6-dimensional nilpotent binary Lie algebras,
all 6-dimensional nilpotent anticommutative algebras
have been found.
A full description of degenerations has been obtained
for 2-dimensional algebras,
for 4-dimensional Lie algebras,
for 4-dimensional Zinbiel and 4-dimensional nilpotent Leibniz algebras in [2]},
for 6-dimensional nilpotent Lie algebras in [3]}, [4]},
for 8-dimensional 2-step nilpotent anticommutative algebras [5]},
and for \((n+1)\) -dimensional \(n\) -Lie algebras.
Our main results related to the geometric classification of cited varieties are summarized below.
